Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C26 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4722 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,3)(4,7,8,6)(9,10,14,16)(11,19,21,15)(12,22,23,17)(13,18,24,20)(25,26)(27,33)(28,36)(29,30)(31,32)(34,38)(35,37)(39,40);(2,3)(6,7)(9,11,14,21)(10,15,16,19)(12,18,23,20)(13,17,24,22)(26,32)(29,39)(33,36)(34,37);(1,3,5,2)(4,7,8,6)(10,16)(13,24)(15,19)(17,22)(25,27,31,28)(26,33,32,36)(29,34,39,37)(30,38,40,35);(1,4,5,8)(2,6,3,7)(9,12,14,23)(10,17,16,22)(11,20,21,18)(13,19,24,15)(25,28,31,27)(26,33,32,36)(29,37,39,34)(30,38,40,35);(9,11)(10,15)(12,20)(13,17)(14,21)(16,19)(18,23)(22,24)(25,29)(26,30)(27,34)(28,37)(31,39)(32,40)(33,38)(35,36);(9,13)(10,18)(11,17)(12,19)(14,24)(15,23)(16,20)(21,22)(25,30)(26,29)(27,35)(28,38)(31,40)(32,39)(33,37)(34,36)Polycyclic presentation:
2^9|a^2=hi,b^2=h,c^2=gh,d^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,f^c=fgi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8109181);